Electricity and Magnetism

Mathematical Tools and Fields

01Series Approximations: Keeping One or Two Terms 02Integrals as Sums: Along a Line and a Curved Path 03Line Integrals: Work as the Integral of F · ds 04Closed Line Integrals: Work Around a Loop 05Double Integrals: Area, Charge and Volume 06Closed Surface Integrals: The Sphere and Its Charge 07Electric Flux Through Surfaces and Volume Integrals Problem set0/10 Problem set 20/10 MIT problem set0/4

01Field Theory Versus Action at a Distance 02Scalar Fields: Contour, Color and Relief Maps 03Vector Fields, Sources, Sinks and Divergence 04Curl: Measuring the Circulation of a Field 05Gravitational, Electric and Magnetic Fields 06Drawing Vector Fields: Arrows, Lines and Grass Seeds Problem set0/10 Problem set 20/10 MIT problem set0/1

Electrostatics

01Coulomb's Law: Force Between Two Point Charges 02Superposition: Adding Coulomb Forces as Vectors 03Field Lines, Null Points and Uniform Fields 04The Electric Dipole and Its 1/r³ Field 05Torque, Energy and Force on a Dipole in a Field Problem set0/10 Problem set 20/10 MIT problem set0/1

01Charge Density and the Field on a Rod's Axis 02The Field on a Charged Rod's Perpendicular Bisector 03The Fields of a Charged Ring and a Charged Disk 04How Field Lines Carry Force, Energy and Momentum 05Electric Pressure and Tension: The Stress ε₀E²/2 Problem set0/10 Problem set 20/10 MIT problem set0/6

Charge Density and the Field on a Rod's Axis

How do charge densities turn a rod's field into an integral, and when does the rod look like a point charge?

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Your summary note

1. Volume, surface and line charge densities $\rho$ , $\sigma$ , $\lambda$
  Define each density as $\rho = \frac{dq}{dV}$ , $\sigma = \frac{dq}{dA}$ , $\lambda = \frac{dq}{d\ell}$ with its SI unit, and write the integral of each density that gives the total charge $Q$ .
2. The field of a continuous distribution as a vector integral
  Write Coulomb's law for one element $dq$ , sum the contributions into $\vec{E} = \frac{1}{4\pi\varepsilon_0}\int \frac{dq}{r^2}\hat{r}$ , and record that this stands for three separate component integrals.
3. Worked example: the field on a uniform rod's axis
  Set up the rod from $x_0$ to $x_0+\ell$ with $dq = \lambda\,dx'$ and $\hat{r} = -\hat{i}$ , integrate to $\vec{E} = -\frac{Q}{4\pi\varepsilon_0 x_0(\ell+x_0)}\hat{i}$ , and recover the point-charge field when $x_0 \gg \ell$ .
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